Path spaces I: A Menger-type result

TL;DR

This paper introduces the concept of path spaces as a combinatorial generalization of finite graphs to infinitary settings and proves a Menger-type theorem for these structures, broadening the scope of graph theory.

Contribution

It presents a new combinatorial framework called path spaces and establishes a Menger-type theorem within this context, extending classical results to infinitary graph-like structures.

Findings

Defined path spaces as a combinatorial generalization of graphs

Proved a Menger-type theorem for path spaces

Applicable to various topological and graph-like structures

Abstract

Infinite graphs are finitary in the sense that their points are connected via finite paths. So what would an infinitary generalization of finite graphs look like? Usually this question is answered with the aid of topology, e.g. in the case of graph-like spaces. Here we introduce a more combinatorial answer, which we call path space, and prove a version of Menger's theorem for it. Since there are many topological path-like objects which induce path spaces, this result can be applied in a variety of settings.